We study the signed valuations of convex semialgebraic sets defined over non-Archimedean fields. This is motivated by the efforts to understand the structure of semialgebraic sets that arise in convex optimization, such as the spectrahedra and the hyperbolicity cones. We give a full characterization of regular sets that are obtained as signed tropicalizations of convex semialgebraic sets, and we prove that the signed tropicalizations of hyperbolicity cones have a more restrictive structure. To obtain our results, we combine two recent advances in the area of tropical geometry: the study of signed valuations of general semialgebraic sets and the separation theorems for signed tropical convexities.